In a bowl, mix the buttermilk with 1 tablespoon of za'atar. Add the chicken thighs, ensuring they are well coated. Cover and marinate for at least 30 minutes, or overnight for maximum flavor.

In a separate bowl, combine the flour, cornstarch, garlic powder, paprika, salt, and black pepper. Mix thoroughly to ensure even seasoning throughout.

Heat the vegetable oil in a deep skillet or fryer over medium-high heat until it reaches 350°F (175°C). You can test the oil's readiness by dropping in a small piece of bread; it should sizzle immediately.

While the oil heats, remove the chicken from the marinade and allow excess buttermilk to drip off. Dredge each piece of chicken in the flour mixture, pressing gently to ensure an even coating.

Carefully place the coated chicken into the hot oil in batches, ensuring not to overcrowd the pan. Fry for 5-7 minutes on each side, or until golden brown and cooked through. Use a meat thermometer to check that the internal temperature reaches 165°F (74°C).

Once cooked, transfer the chicken to a wire rack lined with paper towels to drain excess oil. Repeat with the remaining chicken.

In a small bowl, combine the Greek yogurt, mayonnaise, minced garlic, lemon juice, dill, and remaining tablespoon of za'atar. Mix until smooth to create a delicious sauce.

To assemble the sandwiches, spread the yogurt sauce on the bottom half of each roll. Place a piece of fried chicken on top, followed by pickled turnips or cucumbers and sliced red onion. Top with the other half of the roll and serve immediately.